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<p>This study focuses on the exponentially weighted moving sample covariance matrix (EWMV), investigating its behavior in both null and alternative hypotheses. Under the null hypothesis, assuming normal observations, we establish exponential probability bounds for the largest eigenvalue. Similarly, under the alternative hypothesis with a single spike, we derive corresponding bounds. We extend our findings to sub-Gaussian and heavy-tailed distribution random vectors, providing exponential bounds for the largest eigenvalue in terms of both the weight parameter and the norm of sub-Gaussian random variables. Furthermore, we address the sparse signal case by adapting our techniques as the dimension of observation approaches infinity. This comprehensive exploration improves our understanding of the behavior of the exponentially weighted moving sample covariance matrix in various statistical settings. These results are then used to develop and compare several sequential monitoring charts of change in covariance matrix using the EWMV spectrum. We illustrate the practical utility of our methods with a real-time example involving the monitoring of covariance changes in EEG data during a mental arithmetic task.</p>

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Spectral Norm of Exponentially Weighted Moving Sample Covariance Matrix and Its Application to Sequential Sparse Signal Detection

  • 期刊论文

  • 发表日期：

    2025年3月20日

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  • Yanhong Wu,
  • Wei Biao Wu,
  • Dong-Yun Kim

摘要

This study focuses on the exponentially weighted moving sample covariance matrix (EWMV), investigating its behavior in both null and alternative hypotheses. Under the null hypothesis, assuming normal observations, we establish exponential probability bounds for the largest eigenvalue. Similarly, under the alternative hypothesis with a single spike, we derive corresponding bounds. We extend our findings to sub-Gaussian and heavy-tailed distribution random vectors, providing exponential bounds for the largest eigenvalue in terms of both the weight parameter and the norm of sub-Gaussian random variables. Furthermore, we address the sparse signal case by adapting our techniques as the dimension of observation approaches infinity. This comprehensive exploration improves our understanding of the behavior of the exponentially weighted moving sample covariance matrix in various statistical settings. These results are then used to develop and compare several sequential monitoring charts of change in covariance matrix using the EWMV spectrum. We illustrate the practical utility of our methods with a real-time example involving the monitoring of covariance changes in EEG data during a mental arithmetic task.